Energy function parameterization

Schematic diagram of the potential energy functions parameterized in the above table, from [99Len] (with permission from the authors and from The American Physical Society).

Wesolowski, T. A. Weber, J. Kohn-Sham equations with constrained electron density the effect of various kinetic energy functional parameterizations on the ground-state molecular properties. Int. J. Quantum Chem. 1997,61, 303-311. 

FORDO s are determined by their occupation numbers and their NGSO s, a relationship that is only unique up to unitary transformations that mix NGSO s with the same occupation numbers. However one can parameterize this association to make it unique. Hence on the paths determined by the constrained energy functional, one has a 1-1 correspondence between and densities,... 

Krieger E, Darden T, Nabuurs SB, Finkelstein A, Vriend G. Making optimal use of empirical energy functions force-field parameterization in crystal space. Proteins 2004 57 678-83. 

The parameterized, analytical representations of fi, ., fiy, fifi determined in the fitting are in a form suitable for the calculation of the vibronic transition moments V fi V") (a—O, +1), that enter into the expression for the line strength in equation (21). These matrix elements are computed in a manner analogous to that employed for the matrix elements of the potential energy function in Ref. [1]. 

For this reason considerable effort goes into the development of the parameters which appear in the energy function (2, ). This parameterization is generally accomplished by the matching of calculated properties to experimental measurements, as a function of the parameter set for selected small model compounds. 

Most force fields used in coordination chemistry, in respect of the organic part of the molecules, are based on or are at least similar to the MM2 11 or AMBER 11 parameterization schemes, or mixtures thereof. However, it is of importance to stress again that transferring parameters from one force field to another without appropriate checks is not valid. This is not only a question of the different potential energy functions that may be used, but it is also a consequence of the interrelatedness of the entire set of parameters. Force field parameters imported from any source, whether a well-established force field or experimental data, should only be used as a starting point for further parameter refinement. 

Also, metal ion directed stereoselective syntheses often involve organometallic complexes. While there is no fundamental difference between metal-carbon and metal-heteroatom bonds, modeling rc-bonded ligands is not trivial.1 Given a known reaction mechanism (which is not possible for many catalytic reactions), the main problem is the parameterization of the potential energy functions for the intermediates and transition states. The problem is that force field parameters are generally carefully fitted to experimental results, i.e., structures or other data related to the output of force field calculations of the type of compound to be modeled have to be available. For short-lived transition states this is a considerable problem. 

Collection of numbers that parameterize the potential energy functions. These include the force constants, the ideal distances and angles, and parameters for van der Waals, electrostatic and other terms. Since the force field parameters are dependent on the potential energy functions, the entire set of functions and parameters are sometimes referred to as the force field . 

In an exact representation of the interaction between a solute and a solvent, i.e., solvation, the solvent molecules must be explicitly taken into account. That is, the solvent is described on a microscopic level, where the individual solvent molecules are considered explicitly. The interaction potential between solvent molecules and between solvent molecules and the solute can, in principle, be found by solving the electronic Schrodinger equation for a system consisting of all the involved molecules. Typically, in practice, a more empirical approach is followed where the interaction potential is described by parameterized energy functions. These potential energy functions (often referred to as force fields) are typically parameterized as pairwise atom-atom interactions. 

In practice, empirical or semi-empirical interaction potentials are used. These potential energy functions are often parameterized as pairwise additive atom-atom interactions, i.e., Upj(ri,r2,..., r/v) = JT. u ri j), where the sum runs over all atom-atom distances. An all-atom model usually requires a substantial amount of computation. This may be reduced by estimating the electronic energy via a continuum solvation model like the Onsager reaction-field model, discussed in Section 9.1. 

MacKerell AD Jr., Brooks B, Brooks CL III, et al., eds. CHARMM The Energy Function and Its Parameterization with an Overview of the Program. Chichester, U.K. John Wiley Sons 1998. 

Two further energy functions based on Poisson-Boltzmann electrostatics and atomic solvation parameter (ASP)-based parameterizations of the solvation free energy changes43 were evaluated by Weng et al.57 in the context of side-chain optimization after rigid-body docking. 

In conventional MM the potential energy function (PEF) is parameterized and these optimized parameters are called force-field parameters. Such methods are widely applied in the studies of nucleic acids, proteins, their complexes and other biomolecular systems. A typical, simple force field of a molecular system is defined by the following equation ... 

In these models, the potential energy function is based on the molecular mechanics all-atom force field and includes the bond, angle, dihedral and non-bonded energy terms. The parameterization is based on the statistical analysis of sets of experimental structures. If a variable q describes a degree of freedom in the system (e.g., bond distances, angles, dihedrals) then, P(q), the probability distribution associated with this degree of freedom, is related to the potential of mean force, W(q), by the following equation... 

MacKerell AD, Brooks B et al (1998) CHARMM the energy function and its parameterization with an overview of the program in The Encyclopedia of Computational Chemistry, Wiley, Chichester... 

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